Yvette Cooper, who said a 'one nation immigration policy' had to work for all. Photograph: Stefan Rousseau/PA

The shadow home secretary, Yvette Cooper, has outlined a package of measures to restrict immediate access to welfare benefits for new European migrants coming to Britain that could be swiftly implemented.

In a major speech outlining Labour's new approach to immigration, Cooper said the government was right to look at the area but says that specific practical proposals are needed instead of the current "frenzy of briefing and rhetoric" by ministers.

Labour's decision to more than match the government's attempt to tighten new migrants' access to benefits and public services comes in advance of the complete opening of the British labour market to Bulgarians and Romanians at the end of the year.

Coalition ministers are struggling to agree a clear set of measures. The latest cabinet committee meeting on the issue took place on Wednesday.

The shadow home secretary says that swift action is possible within existing European rules to ban new migrants claiming jobseeker's allowance within a few days or weeks. This could be done by adding a "presence" test to the existing habitual residence test which new migrants currently have to meet to claim unemployment benefits.

Cooper also said it was possible to take action to ensure that migrants were expected to be in the country for some time or to contribute before they got something back.

Labour also wants to open talks in Europe to end a longstanding provision that requires family benefits, such as child benefit and child tax credits, to be paid to family members who live abroad. Cooper said it was unfair that someone could move to London and leave their children in Paris or Prague and claim British family benefits and send them home.

"Most people who come to Britain from Europe work hard and contribute more in taxes than they use in public services or claim in benefits," Cooper said in her speech to the Institute of Public Policy Research.

"But the system needs to be seen to be fair. Giving people the right to work in other countries was never intended to be a way for people to travel and get support from other countries if they weren't planning to contribute.

"So the government is right to look at this area. But so far they have come up with no specific practical proposals and are engaged in a frenzy of briefing and rhetoric instead."

Cooper said that in addition to these immediate proposals Labour also wants to see longer-term reforms to the benefit rules and residence requirements that are in place across Europe so that individual countries' tax and benefits systems were not unfairly used.

"Requiring countries to treat new migrants exactly the same as long-standing residents create a risk that member states simply cut family support, housing or services for all citizens in order to avoid attracting too many migrant workers. That can't be good for anyone. So Europe should look again at the benefit rules and residence requirements that are in place," she said.

She claimed that this contrasted with the approach of David Cameron, who she said wanted to pull out of European co-operation on the social chapter, on crime, policing and police, and immigration data that would only make European migration problems worse.

Cooper repeated Labour's apology for the mistakes she said the party had made in office, which included a failure to introduce more quickly an Australian-style points based system to restrict low-skilled labour from outside Europe. She also repeated that transitional controls should have been retained for Poles and other new east Europeans preventing them working in Britain.

But she also insisted that tougher enforcement action of the national minimum wage and action against the exploitation of migrant workers by employers keen to undercut domestic labour was needed.

Stronger checks were also needed to curb abuses of short-term student visitor visas, but Cooper said the government should stop targeting overseas higher education students.

She said a "one nation immigration policy" had to work for all: "That means an honest and open debate. It means admitting where we got things wrong and changing," she said.

"It means supporting the government where they get things right, but calling them out when they get it badly wrong. It means recognising that diversity makes Britain stronger. It means no rhetorical arms race, just sensible and practical proposals that can make the system better, stronger and fairer for the future."